The read more button will display if you have a fulltext defined hence this code.
<?php if($params->get('itemReadMore') && $item->fulltext): ?>
<a class="moduleItemReadMore" href="<?php echo $item->link; ?>">
<span><?php echo JText::_('K2_READ_MORE'); ?></span>
</a>
<?php endif; ?>
You can change this by removing && $item->fulltext .
To locate the file and to perform these overrides please read this post .
getk2.org/documentation/tutorials/174-templating-with-k2-and-the-concepts-of-sub-templates
The folder you need to override is located at /modules/mod_k2_content/Default/tmpl